Relations and Functions: Sets and Cartesian Products

Fundamental Definitions of Sets

A set is defined as a well-defined collection of objects. Within the context of mathematics, being "well-defined" implies that there is a clear criterion to determine whether any specific object belongs to the collection or not. The elements within a set are typically referred to as its members or objects, and they are organized to form a distinct grouping.

Methods of Representing Sets

There are two primary ways to represent or describe a set as outlined in the material. The first method is the Roaster form, where all elements of the set are listed explicitly, separated by commas, and enclosed within braces. An example of this is the set of vowels in the English alphabet, denoted as $A = \{a, e, i, o, u\}$. The second method is the Set builder form, which characterizes the elements by a common property or rule rather than listing them individually. For the same set of vowels, the Set builder representation is written as $A = \{x : x \text{ is the vowel}\}$.

Characteristics and Equality of Ordered Pairs

An ordered pair is defined as a collection of elements that are separated by a comma and written in parenthesis, represented as $(a, b)$. Unlike a standard set where the order of elements does not matter, the position of elements in an ordered pair is significant. Equality between two ordered pairs can be used to solve algebraic problems. For example, if we are given the relationship $(x + 7, y - 4) = (10, 2)$, we can find the values of $x$ and $y$ by equating the corresponding components of the pairs. By setting the first components equal, we get $x + 7 = 10$, which simplifies to $x = 3$. By setting the second components equal, we get $y - 4 = 2$, which simplifies to $y = 6$.

Definition of the Cartesian Product

The Cartesian product is a fundamental operation between sets. If $A$ and $B$ are two non-empty sets, then the Cartesian product of $A$ and $B$ is the set of all possible ordered pairs where the first element belongs to set $A$ and the second element belongs to set $B$. This product is denoted by the notation $A \times B$. The formal definition in set-builder notation is expressed as $A \times B = \{(a, b) : a \in A, b \in B\}$.